The book contains more than three thousand mathematics problems and covers each topic taught at school. The problems were contributed by 120 of the higher schools of the USSR and all the universities.

The book is divided into, four parts: algebra and trigonometry, fundamentals of analysis, geometry and vector algebra, and the problems and questions set during oral examinations. The authors considered it necessary to include some material relating to complex numbers, combinatorics, the binomial theorem, elementary trigonometric inequalities, and set theory and the method of coordinates. The authors believe that this material will help the readers systematize their knowledge in the principal divisions of mathematics.

In writing the book, the authors have used their experience of examining students in mathematics at higher schools and the preparation of television courses designed to help students revise their knowledge for the entrance examinations to higher educational establishments.

To make it easier for readers to grasp the material, some of the sections have been supplemented with explanatory text. The problems are all answered and some have additional hints or complete solutions.

The more difficult problems are marked with asterisks. Part 4 is entitled “Oral Examination Problems and Questions” and includes samples suggested by the higher schools.

The authors hope that this book will help those who want to enter the various types of higher school, aid the teachers, and be of use to all those who want to deepen and systematize their knowledge of mathematics.

EDITED BY PROF. A.I. PRILEPKO, D.Sc.

Translated from the Russian by Irene Aleksanova

Contents

Preface 5
Part 1 Algebra Trigonometry and Elementary Functions 9
1.1 Problems on Integers Criteria for Divisibility 9
1.2 Real Numbers Transformation of Algebraic Expressions 13
1.3 Mathematical Induction Elements of Combinatorics Binomial Theorem
1.4 Equations and Inequalities of the First and the Second Degree
1.5 Equations of Higher Degrees Rational Inequalities
1.6 Irrational Equations and Inequalities
1.7 Systems of Equations and Inequalities
1.8 The Domain of Definition and the Range of a Function
1.9 Exponential and Logarithmic Equations and Inequalities
1.10 Transformations of Trigonometric Expressions Inverse Trigonometric Functions
1.11 Solution of Trigonometric Equations Inequalities and Systems of Equations
1.12 Progressions
1.13 Solution of Problems on Derivation of Equations
1.14 Complex Numbers
Part 2 Fundamentals of Mathematical Analysis
2.1 Sequences and Their Limits An Infinitely Decreasing Geometric Progression Limits of Functions
2.2 The Derivative Investigating the Behaviour of Functions with the Aid of the Derivative
2.3 Graphs of Functions
2.4 The Antiderivative The Integral The Area of a Curvilinear Trapezoid
Part 3 Geometry and Vector Algebra
3.1 Vector Algebra
3.2 Plane Geometry Problems on Proof
3.3 Plane Geometry Construction Problems
3.4 Plane Geometry Calculation Problems
3.5 Solid Geometry Problems on Proof
3.6 Solid Geometry Calculation Problems
Part 4 Oral Examination Problems and Questions 241
4.1 Sample Examination Papers 241
4.2 Problems Set at an Oral Examination 244
Hints and Answers 265
Appendix 386